It is obvious that \(f(x)\)  is \(x_{0}\) -APN for any \(x_{0}\in \mathbb {F}_{2^{n}}\)  if \(f(x)\)  is an APN function.
Partially APN power functions are of APN-like exponents that are not APN. In [1]} the authors provided some characterizations and propositions of partially APN functions.
Furthermore, they presented several classes of \(x_{0}\) -APN functions for some \(x_{0}\in \mathbb {F}_{2^{n}}\) . Pott [2]} pointed out that for any \(n\ge 3\)  there exist partial 0-APN permutations on \(\mathbb {F}_{2^n}\) , when he studied the relation between Steiner triple systems and partial 0-APN permutations. Based on the idea and the known instances of \(x_{0}\) -APN functions, it is very interesting to construct more infinite classes of \(x_{0}\) -APN functions for some \(x_{0}\in \mathbb {F}_{2^{n}}\) .
If \(f\)  is a power function, then we only need to consider the partial APN properties of \(f\)  at 0 or 1 since it's special algebraic structure [1]}. Moreover, \(f\)  is 0-APN if and only if the equation \(f(x + 1) + f(x) + 1 = 0\)  has no solution in \(\mathbb {F}_{2^n}\backslash \lbrace 0,1\rbrace \) . In [1]}, [5]}, Budaghyan et al. explicitly constructed a number of 0-APN but not APN power functions \(f(x) = x^d\)  over \(\mathbb {F}_{2^{n}}\) . They further listed all power functions over \(\mathbb {F}_{2^{n}}\)  for \(1\le n\le 11\)  that are 0-APN but not APN in Table 1 of [5]}. Recently, Qu and Li [7]} constructed seven classes of 0-APN power functions over \(\mathbb {F}_{2^n}\) . Two of them were proved to be locally-APN [8]}.
The purpose of this paper is to construct more new classes of 0-APN power functions over \(\mathbb {F}_{2^n}\) .
As a result, we propose several new infinite classes of 0-APN power functions over \(\mathbb {F}_{2^n}\)  by using the multivariate method and resultant elimination. For the sake of clarity and further studying, we list the new results of this paper in Table REF , along with some prior ones.
